A Divergence Preserving Cut Finite Element Method for Darcy Flow

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1793-A1820, June 2024.
Abstract. We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs [math], [math]. Here [math] is the space of discontinuous polynomial functions of degree less than or equal to [math] and [math] is the Raviart–Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on [math], [math]. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs [math], [math], and [math] (where [math] is the Brezzi–Douglas–Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/CutFEM/CutFEM-Library and in the supplementary materials (CutFEM-Library-master.zip [30.5MB]).